A 2-L flask contains 1.6 g of methane and 0.5 g of hydrogen at `27^(@)C`. Calculate the partial pressure of each gas in the mixture and hence calculate the total pressure.

Explanation

To solve the problem of calculating the partial pressures of methane and hydrogen in a 2-L flask at 27°C, we will follow these steps:

### Step 1: Calculate the number of moles of each gas.
1. **Moles of Methane (CH₄)**:
- Given mass of methane = 1.6 g
- Molar mass of methane (CH₄) = 12.01 (C) + 4 × 1.008 (H) = 16.04 g/mol
- Moles of CH₄ = mass / molar mass = 1.6 g / 16.04 g/mol ≈ 0.0997 mol

2. **Moles of Hydrogen (H₂)**:
- Given mass of hydrogen = 0.5 g
- Molar mass of hydrogen (H₂) = 2 × 1.008 = 2.016 g/mol
- Moles of H₂ = mass / molar mass = 0.5 g / 2.016 g/mol ≈ 0.248 mol

### Step 2: Calculate the total number of moles in the mixture.
- Total moles (n_total) = Moles of CH₄ + Moles of H₂
- n_total = 0.0997 mol + 0.248 mol ≈ 0.3477 mol

### Step 3: Use the Ideal Gas Law to calculate the partial pressures.
The Ideal Gas Law is given by the equation:
\[ PV = nRT \]
Where:
- P = pressure (in atm)
- V = volume (in L)
- n = number of moles
- R = ideal gas constant = 0.0821 L·atm/(K·mol)
- T = temperature in Kelvin

First, convert the temperature from Celsius to Kelvin:
- T(K) = 27°C + 273.15 = 300.15 K

#### Partial Pressure of Methane (P_CH₄):
Using the Ideal Gas Law:
\[ P_{CH₄} = \frac{n_{CH₄}RT}{V} \]
\[ P_{CH₄} = \frac{(0.0997 \, \text{mol}) \times (0.0821 \, \text{L·atm/(K·mol)}) \times (300.15 \, \text{K})}{2 \, \text{L}} \]
\[ P_{CH₄} ≈ \frac{2.448}{2} ≈ 1.224 \, \text{atm} \]

#### Partial Pressure of Hydrogen (P_H₂):
Using the Ideal Gas Law:
\[ P_{H₂} = \frac{n_{H₂}RT}{V} \]
\[ P_{H₂} = \frac{(0.248 \, \text{mol}) \times (0.0821 \, \text{L·atm/(K·mol)}) \times (300.15 \, \text{K})}{2 \, \text{L}} \]
\[ P_{H₂} ≈ \frac{6.086}{2} ≈ 3.043 \, \text{atm} \]

### Step 4: Calculate the total pressure.
- Total pressure (P_total) = P_CH₄ + P_H₂
- P_total = 1.224 atm + 3.043 atm ≈ 4.267 atm

### Final Answer:
- Partial pressure of methane (P_CH₄) ≈ 1.224 atm
- Partial pressure of hydrogen (P_H₂) ≈ 3.043 atm
- Total pressure (P_total) ≈ 4.267 atm